Topology Essay

2269 words - 9 pages

Topology

Topology is the study of those properties of geometric figures that are unchanged when the shape of the figure is twisted, stretched, shrunk, or otherwise distorted without breaking. It is sometimes referred to as "rubber sheet geometry" (West 577). Topology is a basic and essential part of any post school mathematics curriculum. Johann Benedict Listing introduced this subject, while Euler is regarded as the founder of topology. Mathematicians such as August Ferdinand Möbius, Felix Christian Klein, Camille Marie Ennemond Jordan and others have contributed to this field of mathematics. The Möbius band, Klein bottle, and Jordan curve are all examples of objects commonly studied. These and other topics prove to be intricate and fascinating mathematical themes.

Topologists are mathematicians who study qualitative questions about geometrical structures. They ask questions like does the structure have any holes in it? Is it all connected, or can it be separated into parts? Topologists are not concerned with size, straightness, distance, angle, or other such properties. An often-cited example is the London Underground map. This will not reliably tell you how far it is from Kings Cross to Picadilly, or even the compass direction from one to the other. However, it will tell you how the lines connect between them, using topological rather than geometric information (What 1).

Furthermore, if one figure can be distorted into another figure without breaking, then the two figures are described as being topologically equivalent to each other. Two examples of topologically equivalent figures are a coffee cup and doughnut, and groups of the letters of the alphabet. First, an object shaped like a doughnut is a torus. A torus can be molded into a coffee cup without tearing it or breaking it. Also, the letters C, I, L, M, N, S, U, V, W, and Z are all topologically equivalent. C can be straightened out to make an I, the I can be bent to make an L, and so on. O and D are topologically equivalent only to each other (West 578).

Topology has an interesting history. As a branch of mathematics, it did not spring full-blown into the minds of some mathematicians. It gradually developed as a number of mathematicians experimented with the distortion of geometric figures. In the 18th and 19th centuries, Euler distorted a map into a network and concluded that the formula v - e + f = 2, where v equals the number of vertices, e equals the number of edges, and f equals the number of faces, holds true for all solids. Then, Antoine-Jean Lhuilier attempted to classify cases in which he discovered that Euler's formula was wrong. Also, Möbius discovered a one-sided surface. Carl Friedrich Gauss, a German mathematician, explored the distortion of knots. Listing published his Census; Bernhard Riemann studied the multiplicities of the roots of equations, while Klein and Fricke developed his ideas. Gustav Kirchhoff wrote on the flow of current in the electrical...

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